Trigonometry

y(t) -.65 -.84 -.96 -.99 -.94 -.8 -.58
x(t) -.99 -.89 -.49 0.08 0.63 0.96 0.95

y(t) -.32 -.03 0.27 0.54 0.76 0.92 0.99
x(t) 0.61 0.06 -.51 -.90 -.98 -.72 -.12

y(t) 0.98 0.87 0.69 0.45 0.17 -.12 -.41
x(t) 0.39 0.84 1 0.81 0.33 -.25 -.75

y(t) -.66 -.85 -.97 -.99 -.93 -.79
x(t) -1 -.88 -.46 0.11 0.65 0.96


Now plot the red data against the purple data. Your

graph paper will scale −1 ≤ ≤ 1 in steps of 0.2 and
−1 ≤ ≤ 1 also in steps of 0.2. Join the data points.


























This is an example of a Lissajous Figure.


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Example 15
rd
A 3 polynomial has three roots which are,

tan(31.94), cos(102.85), −sin(64.28)


Determine the coefficients of the polynomial.

Let


= tan(31.94), = cos(102.85), = −sin(64.28)

Then


( − )( − )( − )

Expand,


2
[ − ( + ) + ]( − )
3
2
2
− − ( + ) + ( + ) + −
3
2
− ( + + ) + ( + + ) −
In your fx calculator, store the trigonometric values,
enter the following keystrokes,


l31.94)=q(STO)(A)
k102.85)=q(STO)(B)

zj64.28)=q(STO)(C)





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2
The x coefficient is −( + + ),
Cz(J(A)+J(B)+J(C))=








The x coefficient is + +

C(J(A)J(B)+J(A)J(C)+

J(B)J(C))=







0
The x coefficient is
CzJ(A)J(B)J(C)=









The cubic polynomial is,

2
3
+ 0.4999 − 0.4999 − 0.1249
Divide through by 0.1249 and the polynomial

becomes,

2
3
8 + 4 − 4 − 1
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Example 16
Prove that for any triangle whose angles are A, B and

C,


sin(2 ) + sin(2 ) + sin(2 )
= 2sin( ) sin( ) sin( )


Express as,

= [sin(2 ) + sin(2 )] + sin(2 )


Use the identity,


sin(2 ) + sin(2 ) = 2 sin( + ) cos( − )

Then


= 2 sin( + ) cos( − ) + cos(2 )

But + = 180 −


sin( + ) = sin(180 − )


= sin(180) cos( ) − cos(180) sin( )

= sin( )


= 2 sin( ) cos( − ) + cos(2 )

but


cos(2 ) = 2 sin( ) cos( )

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Leaving

= 2 sin( ) cos( − ) + 2 sin( ) cos( )


= 2 sin( ) [cos( − ) + cos( )]


But = 180 − ( + )

Therefore,


cos( ) = cos(180 − ( + ))

= cos(180) cos( + ) + sin(180) sin( + )


= −cos( + )


Leaving

= 2 sin( ) [cos( − ) − cos( + )]


Now

cos( − ) = cos( ) cos( ) + sin( ) sin( )


cos( + ) = cos( ) cos( ) − sin( ) sin( )

When these are subtracted,


cos( − ) − cos( + ) = 2 sin( ) sin( )

Leaving


= 4 sin( ) sin( ) sin( )



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For further trigonometry methods and worked

examples, refer to the following book from

www.CambridgePaperbacks.com

















































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http://online.anyflip.com/pchj/mncx
/#p=1














http://online.anyflip.com/pchj/fxiq/#
p=1













http://online.anyflip.com/pchj/vrtk/
#p=1










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